Adjoints and canonical forms of tree amplituhedra
DOI:
https://doi.org/10.7146/math.scand.a-149816Abstract
We consider semi-algebraic subsets of the Grassmannian of lines in three-space called tree amplituhedra. These arise in the study of scattering amplitudes from particle physics. Our main result states that tree amplituhedra in $\mathrm {Gr}(2,4)$ are positive geometries. The numerator of their canonical form plays the role of the adjoint in Wachspress geometry, and is uniquely determined by explicit interpolation conditions.
References
Arkani-Hamed, N., Bai, Y., and Lam, T., Positive geometries and canonical forms, J. High Energy Phys. 2017, no. 11, 039, front matter+121 pp. https://doi.org/10.1007/jhep11(2017)039
Arkani-Hamed, N., Hodges, A., and Trnka, J., Positive amplitudes in the amplituhedron, J. High Energy Phys. 2015, no. 8, 030, front matter+24 pp. https://doi.org/10.1007/JHEP08(2015)030
Arkani-Hamed, N., Thomas, H., and Trnka, J., Unwinding the amplituhedron in binary, J. High Energy Phys. 2018, no. 1, 016, front matter+40 pp. https://doi.org/10.1007/jhep01(2018)01
Arkani-Hamed, N., and Trnka, J., The amplituhedron, Journal of High Energy Physics 2014, no. 10, 030, front matter+30 pp. https://doi.org/10.1007/JHEP10(2014)030
Even-Zohar, C., Lakrec, T., Parisi, M., Tessler, R., Sherman-Bennett, M., and Williams, L., A cluster of results on amplituhedron tiles, Lett. Math. Phys. 114 (2024), no. 5, Paper No. 111. https://doi.org/10.1007/s11005-024-01854-4
Franco, S., Galloni, D., Mariotti, A., and Trnka, J., Anatomy of the amplituhedron, J. High Energy Phys. 2015, no. 3, 128, front matter+61 pp. https://doi.org/10.1007/JHEP03(2015)128
Fulton, W., Intersection theory, Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer-Verlag, Berlin, 1998. https://doi.org/10.1007/978-1-4612-1700-8
Gaetz, C., Positive geometries learning seminar, unpublished lecture notes, available at https://math.mit.edu/ tfylam/posgeom/gaetz_notes.pdf (2020).
Galashin, P., Karp, S. N., and Lam, T., The totally nonnegative Grassmannian is a ball, Adv. Math. 397 (2022), Paper No. 108123, 23 pp. https://doi.org/10.1016/j.aim.2021.108123
Kohn, K., Piene, R., Ranestad, K., Rydell, F., Shapiro, B., Sinn, R., Sorea, M.-S., and Telen, S., Adjoints and canonical forms of polypols, arXiv:2108.11747 (2021).
Kohn, K., and Ranestad, K., Projective geometry of Wachspress coordinates, Found. Comput. Math. 20 (2020), no. 5, 1135–1173. https://doi.org/10.1007/s10208-019-09441-z
Lam, T., An invitation to positive geometries, arXiv:2208.05407.
Łukowski, T., On the boundaries of the $m=2$ amplituhedron, Ann. Inst. Henri Poincaré D 9 (2022), no. 3, 525–541. https://doi.org/10.4171/aihpd/124
Łukowski, T., Parisi, M., and Williams, L. K., The positive tropical Grassmannian, the hypersimplex, and the $m= 2$ amplituhedron, Int. Math. Res. Not. IMRN 2023, no. 19, 16778–16836. https://doi.org/10.1093/imrn/rnad010
Mandelshtam, Y., Pavlov, D., and Pratt, E., Combinatorics of $m= 1$ grasstopes, arXiv:2307.09603.
Parisi, M., Sherman-Bennett, M., and Williams, L., The $m=2$ amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers, Commun. Am. Math. Soc. 3 (2023), 329–399. https://doi.org/10.1090/cams/23
Sinn, R., Algebraic boundaries of (mathrm SO(2))-orbitopes, Discrete Comput. Geom. 50 (2013), no. 1, 219–235. https://doi.org/10.1007/s00454-013-9501-5
Sturmfels, B., Totally positive matrices and cyclic polytopes, Linear Algebra Appl. 107 (1988), 275–281. https://doi.org/https://doi.org/10.1016/0024-3795(88)90250-9
